Analytic representation of $F_K/F_\pi$ in two loop chiral perturbation theory

We present an analytic representation of $F_K/F_\pi$ as calculated in three-flavour two-loop chiral perturbation theory, which involves expressing three mass scale sunsets in terms of Kamp\'e de F\'eriet series. We demonstrate how approximations may be made to obtain relatively compact analytic representations. An illustrative set of fits using lattice data is also presented, which shows good agreement with existing fits.


Introduction-
The spectrum of QCD contains as lightest particles the pseudo-scalar octet, and their properties provide a delicate test of its nonperturbative features, including that of chiral symmetry breaking in the sector involving the three lightest quarks. Of these, a special place is accorded to the decay constants of the kaon and pion, namely F K and F π . Their ratio has been investigated on the lattice now, even at quark masses that include the physical values [1]. On the other hand, in chiral perturbation theory (ChPT) [2] at two-loops, expressions have been available for nearly two decades, but involving certain integrals (sunsets) that are evaluated numerically [3]. In this work, we provide an analytic expression for F K /F π , which among other things incorporates double series derived using Mellin-Barnes (MB) representations of the sunsets. This allows us to produce a template for easy fitting to lattice simulations.
Methodology-Three-flavour ChPT expressions for the decay constants of the pseudoscalar mesons at twoloops are given in [3]. These may be decomposed as: CT + (F P ) (6) loop + O(p 8 ), (1) where P is the particle in question. The O(p 6 ) contribution can be subdivided as: d P Li×log collects the terms linear in the O(p 4 ) LECs L i and containing chiral logs, d P log , d P log×log collect the terms linear respectively quadratic in chiral logarithms without L i , d Li and d P Li×Lj the terms linear respectively quadratic in the LECs L i . The term (F P ) (6) CT is composed of the O(p 6 ) counterterms, i.e. the LECs C r i , while d P sunset are the pure sunset terms. One determines the ratio F K /F π using: The terms d P sunset are not available fully analytically. Their determination is the goal of this work. The sunset integral is defined as: Aside from the basic scalar integral defined above, tensor integrals in which the momenta q µ and q µ q ν appear in the numerator, and derivatives with respect to the external momentum of both the scalar and tensor integrals contribute to d P sunset [3]. The tensor integrals, as well as all the derivatives, may be reduced into a linear combination of scalar integrals using the methods given in [4]. Thus only a smaller set of master integrals (MI) is needed. ) and the one-loop tadpole integral. The problem reduces to solving these analytically in the required mass configurations. For the evaluation of F K /F π , seven distinct three mass scale MI need evaluation.
MB theory leads to representations of these MI where each integral consists of at least one double complex plane integral. These double MB integrals are evaluated using the method proposed in [5] and fully systematized in [6] to obtain results in the form of sums of single and double infinite series [7]- [9].
The analytic representation-Using Eq.(3), we obtain the following representation of F K /F π : where ξ π = m 2 π /(16π 2 F 2 π ), ξ K = m 2 K /(16π 2 F 2 π ), λ i = log(m 2 i /µ 2 ), and: F F consists of the terms arising from the pure sunset contributions. The split between theK i terms and F F is not unique: one convenient decomposition, that takes into account the freedom to distribute the chiral logs while keeping the final result unchanged, is: where: The MI are denoted by H , the "bar" indicating that the chiral subtraction prefactor µ 2 e γ E −1 4π 4−d has been taken into acount and that the chiral logarithms have been extracted and included in the log terms of Eq.(2). Expressions for the two mass scale MI are given in [10], and those for the three mass scale are given below in terms of generalized hypergeometric ( p F q ) and Kampé de Fériet (KdF) series. The three mass scale MI not explicitly presented here can be derived from the following by differentiation w.r.t the appropriate square propagator mass. The validity of Eqs.(15)-(17) is dictated by the region of convergence of the KdF and p F q series, which is given by (m π < m η ) ∧ (m π + m η < 2m K ) and shown in Fig. 1.   1 + 2α, 2 + 2α, 3 + 2α : 1, 1 and . (17) One may obtain simplified representations for F F by truncating the series at the desired precision, and taking an expansion around ρ = m 2 π m 2 K = 0. For illustrative purposes, we present one such representation in which we truncate the series such that the error between the exact and truncated values is < 1% for most of the sets of masses used in the lattice study of [1]. We get: The range of validity of Eqs.(18)- (19) is shown in Fig. 2, in which the exact value of F F is plotted against x = √ ρ, as are the approximate F F retained up to various orders of ρ. The expansion up to O(ρ 4 ) approximates the exact value of F F to 1% for m π /m K < 3 and to 6% for m π /m K < 0.5. One may obtain a representation with greater accuracy by truncating the series with a larger number of terms.
For the reader to be able to verify the implementation of these expressions, we give the numerical values of F K /F π coming from both exact and approximate expressions and obtained with physical values m π = 0.1350GeV, m K = 0.4955GeV, F π = 0.0922GeV, as well as the LEC values of the BE14 fit of [11]. We get, using Eq.(8), , Illustrative Lattice Fits-In this section, we present an exploratory numerical study based on our analytical representation by fitting Eq.(5) with the data of the lattice study [1] to determine best-fit values of the NLO LEC L r 5 and the NNLO LEC combinations C r 14 + C r 15 and C r 15 + 2C r 17 . We perform the fit (using [12]) on the mass sets for which m π < 0.40 GeV. We do the fit on the 'exact' F F , i.e. truncating the KdF series after 1000 2 terms, and cross-check by fitting the exact purely numerical version of Eq.(3) with CHIRON [13]. The fit on the approximate version presented in Eq. (18) gives compatible results.
The uncertainties on the values of the LEC given in this section derive from the errors of the F K /F π data of the lattice study, but do not take into account other uncertainties. As detailed in [1], systematic effects due to lattice artificats can arise from correlator fit time choices, lattice spacings, renormalization and finite volume corrections, among other things. When these effects are taken into account, such as by means of the results presented in [14,15] to account for the extrapolation to infinite volume, the values of the LEC presented in this section are likely to change. However, determining the exact nature and magnitude of the change involves a detailed study that is outside the scope of this paper. Therefore, the numerical results in this section are given for an illustrative purpose only, to encourage the lattice community to undertake just such a detailed study using the NNLO analytic results presented above.
We fix the renormalization scale µ at m ρ = 0.77 GeV, and use the values of the BE14 fit [11]  other L r i . In addition we fix F π in the determination of ξ π and ξ K to 92.2 MeV and obtain: The correlation parameters are given in Table I and the quality of the fit is shown in Fig. 3 (Left). The correlation is shown graphically in Fig. 3 (Middle, Right) by plotting a number of random points in a distribution given by the correlation matrix of the fit projected on the two different planes.
The change in the values above arises primarily due to the variation of F π . Keeping F π fixed at 92.2 MeV but with the set of inputs used to calculate Eq.(24) results in changes of ≈ 20%, 35% and 10% in the Eq.(22) values of the L r 5 , C r 14 +C r 15 and C r 15 +2C r 17 , respectively. As the difference in the inputs for Eq.(22) and Eq.(24) is primarily the data from the coarsest lattices, it seems that the lattice data has a significant impact on fitting the LECs.
Conclusions-The ratio F K /F π is a quantity at the heart of chiral symmetry breaking, a fundamental  property of the strong interactions that is measured in ab initio calculations on the lattice. Tuning of the quark masses to physical values is now possible. Thus an analytic expansion for this quantity in masses of the quarks or the mesons is the order of the day. Using modern loop calculation techniques, we have achieved this goal. At present, two-loop precision is sufficient to fit the lattice data; this might change when the lattice precision improves in the future. While there exist three-loop results in two-flavour ChPT [19], in threeflavour ChPT two-loops is the state of the art, making our method and results all the more significant. This work is a product of combining techniques developed independently in various branches of elementary particle physics and field theory, and represents an important advance on the results that appeared nearly two decades ago, when many sunsets were evaluated numerically. We hope this work will pave the way for detailed comparisons of other similar quantities with lattice simulations, and help improve our understanding of both ChPT and lattice studies.